- Products are defined by attributes, each with a number of levels
- Respondents choose from among product alternatives
- Respondent-level preferences are estimated for each attribute level
- Preference estimates are used to make counterfactual predictions in a simulated market
- Market simulators inform new product development, pricing, product line optimization, etc.
\[
U_{hj} = \beta_{h1}x_{j1} + \beta_{h2}x_{j2} + \cdots + \beta_{hk}x_{jk} + \epsilon_{hj}
\]
\[
\color{grey}{U_{hj} = \color{black}{\beta_{h1}}x_{j1} + \color{black}{\beta_{h2}}x_{j2} + \cdots + \color{black}{\beta_{hk}}x_{jk} + \epsilon_{hj}}
\]
\[
\color{grey}{U_{hj} = \beta_{h1}\color{black}{x_{j1}} + \beta_{h2}\color{black}{x_{j2}} + \cdots + \beta_{hk}\color{black}{x_{jk}} + \epsilon_{hj}}
\]
Discrete Choice
For each respondent \(h\) and choice task \(t\) with \(j\) alternatives and \(k\) attribute levels:
\[
\begin{aligned}
y_{ht} & = \text{argmax}(pr(y_{ht})), \ \text{s.t.} \ y_{ht} \in \left\{1, 2, \cdots, J \right\} \\[2mm]
p(y_{htj}) & = {\exp\left(\beta_{h1}x_{tj1} + \beta_{h2}x_{tj2} + \cdots + \beta_{hk}x_{tjk} \right) \over \sum_{j=1}^J \exp\left(\beta_{h1}x_{tj1} + \beta_{h2}x_{tj2} + \cdots + \beta_{hk}x_{tjk} \right)} \\[2mm]
\color{grey}{B} & \hspace{3mm} \color{grey}{\sim MVN\left(\gamma, \Sigma \right)} \\[2mm]
\color{grey}{\gamma} & \hspace{3mm} \color{grey}{\sim \textit{Normal}(0, 1)} \\[2mm]
\color{grey}{\Sigma} & \hspace{3mm} \color{grey}{= \text{diag}(\tau) \ \Omega \ \text{diag}(\tau)} \\[2mm]
\color{grey}{\Omega} & \hspace{3mm} \color{grey}{\sim LKJ(1)} \\[2mm]
\color{grey}{\tau} & \hspace{3mm} \color{grey}{\sim \textit{Half-Normal}(1, 2)}
\end{aligned}
\]
Discrete Choice
For each respondent \(h\) and choice task \(t\) with \(j\) alternatives and \(k\) attribute levels:
\[
\begin{aligned}
\color{grey}{y_{ht}} & \hspace{3mm} \color{grey}{= \text{argmax}(pr(y_{ht})), \ \text{s.t.} \ y_{ht} \in \left\{1, 2, \cdots, J \right\}} \\[2mm]
\color{grey}{p(y_{htj})} & \hspace{3mm} \color{grey}{= {\exp\left(\beta_{h1}x_{tj1} + \beta_{h2}x_{tj2} + \cdots + \beta_{hk}x_{tjk} \right) \over \sum_{j=1}^J \exp\left(\beta_{h1}x_{tj1} + \beta_{h2}x_{tj2} + \cdots + \beta_{hk}x_{tjk} \right)}} \\[2mm]
B & \sim MVN\left(\gamma, \Sigma \right) \\[2mm]
\color{grey}{\gamma} & \hspace{3mm} \color{grey}{\sim \textit{Normal}(0, 1)} \\[2mm]
\color{grey}{\Sigma} & \hspace{3mm} \color{grey}{= \text{diag}(\tau) \ \Omega \ \text{diag}(\tau)} \\[2mm]
\color{grey}{\Omega} & \hspace{3mm} \color{grey}{\sim LKJ(1)} \\[2mm]
\color{grey}{\tau} & \hspace{3mm} \color{grey}{\sim \textit{Half-Normal}(1, 2)}
\end{aligned}
\]
\[
\LARGE{p(\theta | X) \propto p(X | \theta) \ p(\theta)}
\]
\[
\LARGE{p(\theta, \alpha | X) \propto p(X | \theta) \ \color{red}{p(\theta | \alpha)} \ p(\alpha)}
\]
Discrete Choice
For each respondent \(h\) and choice task \(t\) with \(j\) alternatives and \(k\) attribute levels:
\[
\begin{aligned}
\color{grey}{y_{ht}} & \hspace{3mm} \color{grey}{= \text{argmax}(pr(y_{ht})), \ \text{s.t.} \ y_{ht} \in \left\{1, 2, \cdots, J \right\}} \\[2mm]
\color{grey}{p(y_{htj})} & \hspace{3mm} \color{grey}{= {\exp\left(\beta_{h1}x_{tj1} + \beta_{h2}x_{tj2} + \cdots + \beta_{hk}x_{tjk} \right) \over \sum_{j=1}^J \exp\left(\beta_{h1}x_{tj1} + \beta_{h2}x_{tj2} + \cdots + \beta_{hk}x_{tjk} \right)}} \\[2mm]
\color{grey}{B} & \hspace{3mm} \color{grey}{\sim MVN\left(\gamma, \Sigma \right)} \\[2mm]
\gamma & \sim \textit{Normal}(0, 1) \\[2mm]
\Sigma & = \text{diag}(\tau) \ \Omega \ \text{diag}(\tau) \\[2mm]
\Omega & \sim LKJ(1) \\[2mm]
\tau & \sim \textit{Half-Normal}(1, 2)
\end{aligned}
\]
